The Bridge of Love in Vrnjacka Banja, Serbia, is a place where young women traditionally go to affirm their ardor for their lovers. A woman will write the name of her beloved on the lock, attach it to the railing, and then toss the key into the river. From a travel website:

If not for the padlocks that cover its railings, you might not even notice the Bridge of Love in the center of Banja. Though it is just one of 15 bridges in Vrnjacka Banja this bridge with a sad story has become the symbol of the city.

Locals tell the story of Relja and Nada, two young lovers who would meet here every night before WWI. Once the war broke out, Relja, who was an officer in the Royal Army, went off to war and never came back. He moved to Greece, married, and forgot all about Nada. Heartbroken, Nada waited for him on the Bridge of Love until her dying day. To avoid reliving Nada’s bitter love story, local love struck girls started coming to the bridge every night to secretly “lock up” their boyfriends’ hearts with padlocks. They did this with the hope of holding on their love for all eternity.

Recently I saw a fountain in Montevideo, Uruguay, where couples attach padlocks with the same intention too.By the way, Ponte Milvio is where Constantine defeated Maxentius after (according to legend) seeing a big Cross in the sky. I didn't know it still existed!

It's an Italian tradition adopted by much of Europe. It's not an old one however. It was completed fabricated in a book , and then a movie called "I Want You" by Federico Moccia in 2006. There's a NYT article about it: http://www.nytimes.com/2007/08/05/world/europe/05iht-rome.4.6991537.html

That being said, it will surely exist forever because it's an awesome tradtition...

The train bridge over the Rhine in Cologne is covered in locks--me and my partner locked one there and threw the keys away in the river this spring! The Germans love it, but they tell me it's an Italian tradition...

I think I recall a similar tradition at the Great Wall of China - the thought process behind that padlock collection is that if one leaves something behind there, then that person is bound to return to that site again during his or her lifetime.